Weak Solutions and Turbulence in Fluid Dynamics

在流体动力学领域，一个主要目标是获得对湍流的详细和定量的了解，湍流是一种在日常使用流体时经常观察到的流体运动，其特征是速度的快速和不规则波动。湍流在无数的工程和工业应用中发挥着关键作用，例如陆地和飞机的设计，在这些应用中，必须深入了解湍流的产生，才能实现节能和有效的技术。长期以来，人们一直认为，湍流的内部工作可以通过用于模拟流体运动的微分方程组来理解，包括欧拉方程。然而，这些方程的分析被证明是如此困难，以至于湍流和流体动力学方程的解之间的大部分关系仍然是猜测和未解决的。最近，通过研究人员和其他数学家的工作，出现了一种新的方法来分析流体方程的解，这些方程的快速波动可以与湍流中看到的相媲美。这种方法与几何数学学科中的问题密切相关。这一研究项目旨在将这一新方法推向其最大的潜力，以提高对湍流的理解，并扩大通过数学分析可以实现的前沿。在技术层面，PI将在流体动力学方程(包括不可压缩的欧拉方程)的“弱解”结构方面证明严格的结果。这些结果将说明这种弱解在多大程度上能够显示出观察到的或预测到的湍流性质。一个研究方向将涉及能量的异常耗散现象，以及这种现象可能存在的程度以及在扰动下的一般性或稳定性。PI将考虑Onsager关于反常耗散猜想的类似和推广，并将研究Euler方程和相关方程一般弱解的动力学结构的精细性质。该研究将实现和扩展凸积分方法，并根据需要引入新的几何或调和分析工具。

In the field of fluid dynamics, a major goal is to obtain a detailed and quantitative understanding of turbulence, which is a type of fluid motion commonly observed in everyday experience with fluids that is characterized by rapid and irregular fluctuations in velocity. Turbulence plays a pivotal role in countless engineering and industrial applications such as the design of land and aircraft, where the production of turbulence must be understood deeply to achieve energy efficient and effective technology. It has long been believed that the inner workings of turbulence may be understood in terms of the differential equations that are being used to model fluid motion, including the Euler equations. However, the analysis of these equations has proved so difficult that much of the relationship between turbulence and solutions to the equations of fluid dynamics remains conjectural and unresolved. Recently through the work of the investigator and other mathematicians, a new approach has emerged to analyze solutions to fluid equations that have rapid fluctuations comparable to what is seen in turbulence. This approach is intimately tied to questions in the mathematical discipline of geometry. This research project aims to push this new approach towards its fullest potential to improve the understanding of turbulence and to expand the frontiers of what can be accomplished through mathematical analysis.At a technical level, the PI will prove rigorous results on the structure of "weak solutions" to the equations of fluid dynamics, including the incompressible Euler equations. The results will address the extent to which such weak solutions can be shown to exhibit the properties observed or predicted of turbulent flows. One direction of research will relate to the phenomenon of anomalous dissipation of energy and the extent to which this phenomenon may exist and be generic or stable under perturbation. The PI will consider analogues and extensions of Onsager's conjecture on anomalous dissipation, and will also investigate fine properties of the dynamical structure of general weak solutions to the Euler and related equations. The research will implement and expand upon the method of convex integration, as well as introduce new tools of geometric or harmonic analysis as needed.